@MISC{Pasupathy_generatinghomogeneous, author = {Raghu Pasupathy}, title = {Generating Homogeneous Poisson Processes}, year = {} }

Share

OpenURL

Abstract

We present an overview of existing methods to generate pseudorandom numbers from homogeneous Poisson processes. We provide three well-known definitions of the homogeneous Poisson process, present results that form the basis of various existing generation algorithms, and provide algorithm listings. With the intent of guiding users seeking an appropriate algorithm for a given setting, we note the computationally burdensome operations within each algorithm. Our treatment includes one-dimensional and two-dimensional homogeneous Poisson processes. Key words: statistics; simulation; random process generation; Poisson processes. Recall that a counting process {Nt, t ≥ 0} is a stochastic process defined on a sample space Ω such that for each ω ∈ Ω, the function Nt(ω) is a “realization ” of the number of “events ” happening in the interval (0, t], with N0(ω) = 0. By this definition, Nt(ω) is automatically integer valued, nondecreasing, and right-continuous for each ω. A homogeneous Poisson process is a type of counting process that is characterized as follows. Definition 1. A counting process {Nt, t ≥ 0} is called a homogeneous Poisson process if: (i) ∀t, s ≥ 0, and 0 ≤ u ≤ t, Nt+s − Nt is independent of Nu; (ii) ∀t, s ≥ 0,Pr{Nt+s − Nt ≥ 2} = o(s); and